What is Algebra?

By Mike Turnbull

Algebra is that branch of mathematics that studies the mathematical relations of abstract representations of numerical values rather than the numerical values per se. These abstract representations form algebraic systems. Algebra is, in general, the practice of manipulating the symbols that represent numerical values in accordance with consistent, well defined, arithmetic rules and conventions.

The symbols used to represent algebraic values can denote either known or unknown numerical values. The known values may indeed be represented as the numerical values themselves¹ (that is, as numbers). The unknown values may be represented by symbols that represent values that do not vary throughout the arithmetic process under consideration - called constant values, or simply constants; or they may be symbols representing unknown values that may take variable values during the arithmetic process - known as variable values, or simply as variables.

The algebraic symbols are formulated into statements, expressions, and equations.

An algebraic statement is a sequence of numerical values or symbols which can be evaluated to a known or unknown value. An algebraic statement is articulated in algebraic symbols and may consist of the bare declaration of a numerical value or a symbol representing a known or unknown value, either variable or constant, or it may be an algebraic expression.

An algebraic expression is a statement with a minimum of two numbers, constants, or variables and at least one mathematical operation either expressed or implied. These mathematical operations can be simple operations such as addition, subtraction, multiplication, or division, or they may be compound operations such as power, surd, logarithm, exponentiation, or any other well defined mathematical operation.

An algebraic equation is a statement that asserts that one algebraic expression evaluates to the same value as another (usually different) expression.

An algebraic inequation is a statement that asserts that one algebraic expression evaluates to a value that is not equal to another (usually different) expression. One expression may evaluate to a value that is less than or greater than that of the other expression².

Algebra is a generalization of arithmetic that includes symbols representing variables and constants the numerical values of which are either known or unknown, as well as known absolute values.

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¹ It must be acknowledged that even absolute numerical values are represented by symbols that have been mutually agreed to signify the absolute numerical value they represent. The symbol ‘2’ is in fact not an absolute numerical value, it is simply a symbol that we agree to regard as a substitute for that underlying absolute numerical value. Any other symbol could have been chosen – in fact, in different cultures, and at different times in human history, other symbols have indeed been used to represent common absolute numerical values.

²The concept of numerical value is not addressed here and it is assumed that the reader has an intuitive understanding of the concept.